Singular Integral Operators, Quantitative Flatness, and Boundary Problems

Description

This monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems – as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis – will find this text to be a valuable addition to the mathematical literature. Preface Acknowledgments Contents 1 Introduction 2 Geometric Measure Theory 2.1 Classes of Euclidean Sets of Locally Finite Perimeter 2.2 Reifenberg Flat Domains 2.3 Chord-Arc Curves in the Plane 2.4 The Class of Delta-Flat Ahlfors Regular Domains 2.5 The Decomposition Theorem 2.6 Chord-Arc Domains in the Plane 2.7 Dyadic Grids and Muckenhoupt Weights on Ahlfors Regular Sets 2.8 Sobolev Spaces on Ahlfors Regular Sets 3 Calderón–Zygmund Theory for Boundary Layers in UR Domains 3.1 Boundary Layer Potentials: The Setup 3.2 SIOs on Muckenhoupt Weighted Lebesgue and Sobolev Spaces 3.3 Distinguished Coefficient Tensors 4 Boundedness and Invertibility of Layer Potential Operators 4.1 Estimates for Euclidean Singular Integral Operators 4.2 Estimates for Certain Classes of Singular Integrals on UR Sets 4.3 Norm Estimates and Invertibility Results for Double Layers 4.4 Invertibility on Muckenhoupt Weighted Homogeneous Sobolev Spaces 4.5 Another Look at Double Layers for the Two-Dimensional Lamé System 5 Controlling the BMO Semi-Norm of the Unit Normal 5.1 Clifford Algebras and Cauchy–Clifford Operators 5.2 Estimating the BMO Semi-Norm of the Unit Normal 5.3 Using Riesz Transforms to Quantify Flatness 5.4 Using Riesz Transforms to Characterize Muckenhoupt Weights 6 Boundary Value Problems in Muckenhoupt Weighted Spaces 6.1 The Dirichlet Problem in Weighted Lebesgue Spaces 6.2 The Regularity Problem in Weighted Sobolev Spaces 6.3 The Neumann Problem in Weighted Lebesgue Spaces 6.4 The Transmission Problem in Weighted Lebesgue Spaces 7 Singular Integrals and Boundary Problems in Morrey and Block Spaces 7.1 Boundary Layer Potentials on Morrey and Block Spaces 7.2 Inverting Double Layer Operators on Morrey and Block Spaces 7.3 Invertibility on Morrey/Block-Based HomogeneousSobolev Spaces 7.4 Characterizing Flatness in Terms of Morrey and Block Spaces 7.5 Boundary Value Problems in Morrey and Block Spaces 8 Singular Integrals and Boundary Problems in Weighted Banach Function Spaces 8.1 Basic Properties and Extrapolation in Banach Function Spaces 8.2 Boundary Layer Potentials on Weighted Banach Function Spaces 8.3 Inverting Double Layer Operators on Weighted Banach Function Spaces 8.4 Invertibility on Homogeneous Weighted Banach Function-Based Sobolev Spaces 8.5 Characterizing Flatness in Terms of Weighted Banach Functions Spaces 8.6 Boundary Value Problems in Weighted Banach Function Spaces 8.7 Examples of Weighted Banach Function Spaces 8.7.1 Unweighted Banach Function Spaces 8.7.2 Rearrangement Invariant Banach Function Spaces References Subject Index Symbol Index